In a railroad system, especially a high-density railway system such as a subway system, vehicles in a train run along a route according to a schedule that can have different travel times that arise from an overall schedule for the high-density railway system. For the travel times, it is necessary to determine an optimal velocity profile for the train, such that energy consumption is minimized, while simultaneously satisfying all constraints of motion, such as velocity limits, safety zones, and etc. More efficient nm-curves for trains and other vehicles can reduce energy consumption.
In the railroad system, the trains can be equipped with regenerative brakes, batteries, and other traction and energy transformation devices. A geometry of the route between stations (locations) is fixed. The geometry indicates the profile of the route, e.g., length, curves, and slope. The resistance from air and tracks are also considered to be a function of the velocity and location of the train along the route. The mass of the train is assumed to be constant, ignoring relatively small variations in the number of passengers and the amount of cargo.
Since travel time requirement is affected by not only the predetermined time-table but also the dynamic situation, the requirement can not be known until just before departure, particularly in high-density railway systems.
At the same time, loading and unloading time can vary dynamically from station to station, depending on time of day, and day of the week. Also, tracks along the route can be under repair during operation of the high-density railway system. All of these conditions lead to changing travel time requirements before the departure time for each trip.
Thus, it is important to optimize the run curves in a short time according to given travel time requirements that are subject to changes before departure.
Dynamics of the system can be described by
                                                        ⅆ              v                                      ⅆ              t                                =                      a            ⁡                          [                                                z                  ⁡                                      (                    t                    )                                                  ,                                  v                  ⁡                                      (                    t                    )                                                  ,                                  u                  ⁡                                      (                    t                    )                                                              ]                                      ,                            (        1        )                                                                    ⅆ              z                                      ⅆ              t                                =                      v            ⁡                          (              t              )                                      ,                            (        2        )            where z(t) represents the location of the vehicle at a time t, v(t) represents the velocity of the vehicle at time t, u(t) represents an action (acceleration, deceleration, braking, coasting, and etc.) taken by the vehicle at time t, and a(z(t), v(t), u(t)) are functions that denote acceleration under the current location of the vehicle, velocity, and action considering various physical factors, e.g., air resistance, track resistance, track slope, motor efficiency, brake efficiency, etc.
A rate of energy consumption E for a vehicle and route is
                              E          =                                    ∫              0              T                        ⁢                                          p                ⁡                                  [                                                            z                      ⁡                                              (                        t                        )                                                              ,                                          v                      ⁡                                              (                        t                        )                                                              ,                                          u                      ⁡                                              (                        t                        )                                                                              ]                                            ⁢                              ⅆ                t                                                    ,                            (        3        )            where T is the travel time.
A power consumption at time t with corresponding vehicle location, velocity, and depends on p(z(t), v(t), u(t)).
Run-curve optimization is a minimization problem with an objective functionJ=μE+(1−μ)T  (4),and the constraints in equations (1), (2), and (3), where a weight μ describes a relative importance of minimizing time vs. energy.
A number of methods for solving this optimization problem are known, such as dynamic programming, heuristic optimization, and nonlinear optimization. K. K. Wong et al (2004) designed heuristics based on nonlinear optimization techniques for solving train run curve optimization problem, where the major efforts are on find optimal coasting-points. Y. Ding et al (2011) also designed a method for computing good costing points using Genetic Algorithms. These heuristic methods can find good but not optimal run curves. At the same time, the computation time increases dramatically as the number of coasting points increases. H. Ko et al (2006) and L. Li et al (2011) developed dynamic programming based algorithm for calculating the optimal run curve for given travel time requirement. These two methods can find the optimal run curves. However, these two methods need large memory storage and long computation time. At the same time, the computational process can not benefit from previous computation. Thus, they are suitable for off-line computation but not able to quickly adapt to newly updated travel time requirement. Our invention can not only compute the optimal run-curve with smaller amount of memory, but also quickly re-compute optimal run-curves for updated travel time requirement by re-using existing computation results.
[1] H. Ko, T. Koseki, and M. Miyatake. Numerical study on dynamic programming applied to optimization of running profile of a train. WIT Press, 103-112, 2004.
[2] L. Li, W. Dong, Y. Ji, and Z. Zhang. An Optimal driving strategy for high-speed electric train. In 2011 30th Chinese Control Conference, pages 5899-5904,2011. [3] Y. Ding, H. Liu, Y. Bai, and F. Zhou. A two-level optimization model and algorithm for energy-efficient urban train operation. Journal of Transportation Systems Engineering and Information Technology, 11(1):96-101,2011. [4] K. K. Wong and T. K. Ho. Coast control for mass rapid transit railways with searching methods. In IEE Proceedings on Electric Power Applications, 2004.